SERIES REPRESENTATIONS AND SIMULATION OF ISOTROPIC RANDOM FIELDS IN THE EUCLIDEAN SPACE

This document represents the Pytho codes associated with Series Representations and Simulations of Isotropic Random Fields in the Euclidean Space by Zhengwei Ma and Chunsheng Ma

Setup

Here we set some initial values for simulation, as well as create some helper functions

(i) Example 3.3 (continued), Case I: \nu=0

We want \begin{equation} y_{1} = -log(U_{0, 1}) \end{equation}

\begin{equation} y_{2} = U_{log(2), log(8)} \end{equation}\begin{equation} v_{1} = y_{1}^{0.5} * exp(\frac{-y_{2}}{2}) \end{equation}\begin{equation} w_{1} = U_{0, 1} \end{equation}\begin{equation} u_{1} = U_{0, 1} \end{equation}

(ii) Example 3.3 (continued), Case II: 0 < \nu < 1

We want \begin{equation} y_{1} = \Gamma(0.5, 1) \end{equation}

\begin{equation} y_{2} = U_{2^{0.5}, 8^{0.5}} \end{equation}\begin{equation} v_{1} = \frac{y_{1}^{0.5}}{y_{2}} \end{equation}\begin{equation} w_{1} = U_{0, 1} \end{equation}\begin{equation} u_{1} = U_{0, 1} \end{equation}

(iii) Example 3.3 (continued), Case III: \nu = 1 or \nu > 1

We want \begin{equation} y_{1} = \Gamma(3, 1) \end{equation}

\begin{equation} y_{2} = U_{\frac{1}{64}, \frac{1}{4}} \end{equation}\begin{equation} v_{1} = y_{1}^{0.5} * y_{2}^{0.25} \end{equation}\begin{equation} w_{1} = U_{0, 1} \end{equation}\begin{equation} u_{1} = U_{0, 1} \end{equation}

(iv) Example 3.5 (continued)

(v) An example from Table 1

For simulation of

\begin{equation} Z(x) = \sqrt(2) \sum_{n=0}^{\infty}J_{n}(rV_{n})cos(n\pi + 2\pi U_{n}) \end{equation}

We being with simulating uniform random variables \begin{equation} U_{0}, U_{1}, U_{2} \sim [0,1]\end{equation}

The distribution function $ F(u) $ has to be chosen from say Table 1.

Consider the function on the second row of Table 1. When $d=2, \alpha=1$ we have \begin{equation} f(u) = \frac{1}{2} * u * exp(\frac{-u^2}{4}), u > 0 \end{equation} and \begin{equation} F(u) = 1 - exp(\frac{-u^2}{4}), u > 0 \end{equation}

Thus, we can use the inverse transformation method to generate random numbers from $F(u)$, where the inverse function of $F(u)$ is \begin{equation} F^{-1}(u) = 2 * \sqrt{-\log(1-u)} \end{equation}

For a uniform random variable $U$ on (0, 1), $U$ and $1-U$ have the same distribution. Thus, a code to generate one $F(u)$ random number is

\begin{equation}v = 2 * \sqrt{\log{U_{0}}} \end{equation}

Demonstrate Z(x) in formula 3.7 with an example, with n=2

We show that the following formula is equivalent